14 LOCALLY CONVEX SPACES 63 For it is clear thatX^*) = MX*); also we have pa(x + y) g pa(jc) + pa(y) £ p(x) + p(y) for all a E I, hence X* + y) ^ p(jc) + p(y). (12.14.2) Let p be a seminorm on a topological vector space E. Then the following conditions are equivalent: (a) p is continuous; (b) there exists a neighborhood V ofOon which p is bounded. Clearly (a) implies (b). Conversely, if p(x) g a for all xe V, then for all ;c0 e E and all x e x0 + fiV we have \p(x) - X*o)l = X* — *o) = a£> an(* there- fore p is continuous on E. (12.14.3) Let (px)a e i 6e any family ofseminorms on a vector space E. Then the topology defined by the pseudo-distances da(x,y) — pjix — y) (12.4) is com- patible with the vector space structure ofE. In view of the definitions, the proofs are the same as in (4.1.1) and (5.1.5). The topology defined by the pseudo-distances d^ is said to be defined by the family ofseminorms (/?«)«6i. Two families of seminorms on E are said to be equivalent if they define the same topology (cf. (12.14.12)). A topological vector space whose topology can be defined by a family ofseminorms is said to be locally convex. If (/>a)aei *s a farcnty ofseminorms defining the topology of a locally convex space E, we obtain a fundamental system of neighborhoods of 0 in E by considering the sets (12.14.3.1) W((oO; r) - {x e E \pai(x) <r for l£i£n} where (a i)^^,, runs through the set of finite families of indices belonging to I, and r is any real number >0. (12.14.4) Let E be a vector space and let (/?a)a e j be a family ofseminorms on E. For the topology defined by the family (pa) to be Hausdorff, it is necessary and sufficient that for each x ^ 0 in E there should exist an index a e I such that P*(X) * 0. This follows from (12.4.4). Hence a seminorm p on a vector space E is a norm if and only if the topology defined by this single seminorm is Hausdorff.is there-